Q Analogs of the binomial coefficient Congruences of Babbage Wolstenholme and Glaisher

1. Q ANALOGS OF THE BINOMIAL COEFFICIENT
CONGRUENCES OF BABBAGE
WOLSTENHOLME AND GLAISHER
MANOJ MOTAPALUKULA
20951A0585

2. INTRODUCTION
In 1819, inspired by Wilson's Theorem which asserts that if p is a prime
(p- 1)! = -1(mod p)---------------- (1.1)
Charles Babbage [2] proved for odd primes p that
( 2P- 1 P – 1)= 1( (mod p2)---------- (1.2)
This is apparently the only paper on number theory written by Babbage,
the famous pioneer of computing machines. His proof is a nice application
of the Chu-Vandermonde summation. In 1862, the Reverend
Wolstenholme [21] improved Babbage's theorem by proving for primes p >
3
(2P- 1 P-1) = 1(mod p3)----------------- (1.3)
Gow [13] presented a biographical note on Wolstenholme

3. Finally, Glaisher in a series of wonderful papers [8-11] proved a large number of
results of this nature,
for example [11, p. 110] if p is a prime > 3, then
(mp+p-1 p- 1)= 1 (mod p3)----------- (1.4) or as he stated the result
(mp+l)(mp+2)...(mp+p-1)=-(p-1)! (mod p3).------------ (1.5)
From (1.5), Glaisher noted that a variety of congruences for binomial coefficients
could be obtained.
we shall prove the q-analog of
(1.5)
(mp+l)(mp+2)...(mp+p-1)=-(p-1)!
(mod p3)

4. THEOREM 1

5. Q-BINOMIAL CONGRUENCES
The most natural consequences of the results in theorem 1 are actually direct
q-generalizations of Babbage's Theorem (1.1).

7. THEOREM 3

9. WOLSTENHOLME'S BINOMIAL
CONGRUENCE
To prove (1.3) with p > 3 Glaisher [11] only needs to invoke (1.5) with m = 1.
Hence to show that we really have a q-generalization of Wolstenholme's
theorem in Theorem 1 we need only show that (1.6) implies (1.5). Now (1.6) is
equivalent to the assertion that there exists a polynomial F(q) with integer
coefficients such that

11. which is divisible by p if N~< p- 2. Therefore the expression in (4.4) is an integer
divisible by p. Hence equating (4.4) and (4.2) we see that (mp + 1)(mp + 2)--.(mp
+ p- 1)- (p- 1)! is divisible by p3. Consequently (1.5) is proved, and thus
Wolstenholme's congruence follows from Theorem

12. WOLSTENHOLME'S HARMONIC SERIES
CONGRUENCE
In order to prove (1.3), Wolstenholme [21] proves auxiliary
congruences. Indeed one of these is much more famous than any of
the congruences previously cited, namely for each prime p > 3
Using Theorem 1, we can prove a related congruence for the q-
harmonic series

14. CONCLUSION
It should be noted that Babbage's original proof of (1.2) can be
directly extended to prove the q-analog. Namely

16. Thank you